APOLONIO DE PERGA Trabajos Secciones cónicas. hipótesis de las órbitas excéntricas o teoría de los epiciclos. Propuso y resolvió el. Nació Alrededor Del Apolonio de Perga. Uploaded by Eric Watson . El libro número 8 de “Secciones Cónicas” está perdido, mientras que los libros del 5. In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the Greek mathematicians with this work culminating around BC, when Apollonius of Perga undertook a systematic study of their properties.
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Science in medieval Islam: What should be considered as a degenerate case of a conic depends on the definition apolonjo used and the geometric setting for the conic section.
It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic.
Furthermore, the four base points determine three line pairs degenerate conics through the base points, each line of the pair containing exactly two base points and so each pencil of conics will contain at most three degenerate conics. This can be done for arbitrary projective planesbut to obtain the real projective plane as the extended Euclidean plane, some specific choices have to be made. This special case is called a rectangular or equilateral hyperbola.
Other sections in this case are called cylindric sections. Further unification is possible if one allows complex numbers as coefficients. The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections.
Euclid was an important representative in the history of mathematics with his work The treaty of the elements, which influenced the development of geometry. Views Read View source View history.
The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga conicass c. This symbolic representation can be made concrete with conicad slight abuse seccilnes notation using the same notation to denote the object as well as the equation defining the object.
Since five points determine a conic, a circle which may be degenerate is determined by three points. An instrument conocas drawing conic sections was first described in CE by the Islamic mathematician Al-Kuhi. These standard forms can be written parametrically as.
Thus, the numbers beyond two or three, had no name, so they used some expression equivalent to “many” to refer to a set older. Apollonius’s study of the properties of these curves made it possible to show that any plane cutting a fixed double cone two nappedregardless of its angle, will produce a conic according to the earlier definition, leading to the secciiones commonly used today. As multiplying all six coefficients by the same non-zero scalar yields an equation with the same set of zeros, one can consider conics, represented by ABCDEF as points in the five-dimensional projective space P 5.
In the complex projective plane the non-degenerate conics can not be distinguished from one another. Although possessed some ability to estimate sizes and magnitudes, they not initially had a notion of number.
In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone.
A conic is the curve obtained as the intersection of a planecalled the cutting planewith the surface of a double cone a cone with two nappes.
Treatise on conic sections
All the conic sections share a reflection property that can be stated as: If the determinant of the matrix of the conic section is zero, the conic section is degenerate. The polar form of the equation of a conic is often used in dynamics ; for instance, determining the orbits of objects revolving about the Sun. Another method, based on Steiner’s construction and which is useful in engineering applications, is the parallelogram methodwhere a conic is constructed point by point by means of connecting certain equally spaced points on a horizontal line and a vertical line.
The type of conic is determined by the value of the eccentricity. A conic section is the locus of all points P whose distance to a fixed point F called the focus of the conic is a constant multiple called the eccentricitye of the distance from P to a fixed line L called the directrix of the conic. Circles, not constructible by the earlier method, are also obtainable in this way. The three types of conic section are the hyperbolathe parabolaand the ellipse.
The theorem also holds for degenerate conics consisting of two lines, but in that case it is known as Pappus’s theorem. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. Many of these have been used as the basis for a definition of the conic sections. Apollonius’s work was translated into Arabic and much of his work only survives through the Arabic version.
In homogeneous coordinates a conic section can be represented as:. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going toward or coming from the second focus are parallel.
The most general equation is of the form . The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane that is its directrix. Dee great innovation of Diophantus is still keeping the algebraic statements rhetoric form of sentence structure, replaced with a series of magnitudes abbreviations, concepts and frequent operators, ie, starts the “syncopated algebra”.
Any point in the plane is on either zero, one or two tangent lines of a conic. It can also be shown : The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone — for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. It shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic. Acnode Crunode Cusp Delta invariant Tacnode. When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric i.
Conic section – Wikipedia
Non-degenerate conic sections are always ” smooth “. One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables or equivalently, the zeros of an irreducible quadratic form.
The three types are then determined by how this line at infinity intersects the conic in the projective space.